Decomposition and stability of linear singularly perturbed systems with two small parameters

In the domain $\Omega =\left\{\left(t,\varepsilon _{1}, \varepsilon _{2} \right): t\in {\mathbb R},\varepsilon _{1}>0, >0\right\}$, we consider a linear singularly perturbed system with two small parameters \[ \left\{ \begin{array}{l} {\dot{x}_{0} =A_{00} x_{0} +A_{01} x_{1} +A_{02} x_{2},} \\ {\varepsilon _{1} \dot{x}_{1} =A_{10} +A_{11} +A_{12} \dot{x}_{2} =A_{20} +A_{21} +A_{22} \end{array}\right. \] where $x_{0} \in R}^{n_{0}}$, $x_{1} R}^{n_{1}}$, $x_{2} R}^{n_{2}}$. this paper, schemes of decomposition and splitting into independent subsystems by using integral manifolds method fast slow variables are investigated. We give conditions under which reduction principle is truthful to study stability zero solution original system.